The generating function is algebraic: closed forms and the growth constant 32/3 for the extension closed subcategories of a uniformly oriented Aₙ₊₁ quiver

The 10ⁿ lower bound Microsoft Copilot asserted and the source could not verify: Rₙ >= 10ⁿ / 111 for every n, at truncation level L = 159

L = 159 is the minimal truncation level at which base 10 is reachable (the source's remark quotes L = 160)

111 is the least constant for base 10: 110 * R₃6 < 10³6

Rₙ >= (10.1)ⁿ / 165 for every n, at the minimal truncation level L = 195, with 165 least (164 fails at n = 42)

Rₙ >= (10.2)ⁿ / 267 for every n, at the minimal truncation level L = 247, with 267 least (266 fails at n = 53)

Where the method stops: F_L(11) > 0 at every truncation level L <= 247, and 111 * R₁2 < 11¹2

The source's own Theorem: Rₙ >= 9ⁿ / 12 at its truncation level L = 47, reproduced; and its constant 12 is least (11 * R₁1 < 9¹1)

Non-vacuity: n -> 2 * 11ⁿ satisfies every hypothesis of the main theorem, and the tabulated Rₙ satisfy the source's truncated inequality for 161 <= n <= 248

Provenance of the tables: the source's a/b recurrence run inside Lean to n = 248 generates both W₀..W₂48 and R₀..R₂48; all 21 OEIS A393920 terms, all 48 Wₜ printed in the source and all 10 OEIS A234268 terms are reproduced exactly

The tables agree with the source's DEFINITION, not only its recurrence: brute force over Condition (star) reproduces R₀..R₅ = 2,7,34,199,1308,9300 and P₀..P₅ = 1,4,21,129,876,6376

The maximal root of the source's own F₄7, of which the PROOF of its Corollary 21 says (not the statement – corrected 2026-08-28) 'we do not know exactly what it is': 9.0116 <= rho₄7 < 9.0117; hence Rₙ >= (9.0116)ⁿ / 12 at the source's own truncation level L = 47 and with its own constant 12, which is least there too (11 fails at n = 10)

The lossless ladder: the source's own proof contains the exact recurrence Rₙ = 2Rₙ₋₁ + sum_(t<n) (Wₜ+Pₜ) Rₙ₋ₜ₋₂ (kernel-checked on the whole table), and its truncations strictly lower every minimal level of the family's ladder — base 9 at L = 45 (source: 47), base 10 at L = 155 (was 159), 10.1 at L = 191 (was 195), 10.2 at L = 242 (was 247) — with the same least constants 12, 111, 165, 267

The ceiling does not move: bases 10.3 and 11 are unreachable at every level L ≤ 248 even on the lossless ladder (G_L ≤ F_L, so each failure implies the W-ladder one) — the convolution the source discards was not the obstruction; externally the same holds to L = 319 and the max reachable base moves only 10.2017 → 10.2098 at L = 248

The first explicit growth bound for OEIS A234268 (convex topologies on an (n+1)-point chain; Tamari interval preposets): Pₙ ≥ 9ⁿ/16, Pₙ ≥ 10ⁿ/144, Pₙ ≥ (10.2)ⁿ/344, each constant least (15/143/343 fail at n = 10/36/53), via the source's Lemma P-exact truncated — not via its §7.2 sandwich, which would give 24/222/534

Controls for the lossless and P packages: explicit witnesses satisfying every hypothesis (Rs = Ps = 2·11ⁿ), and the real tables satisfy the truncated hypotheses at the checkable levels

The P table: P₀…P₂48 (A234268 offset by one) held as literals and pinned to Wfull/Rdata in the kernel by the source's Lemma P-exact; the 21 values of the source's Figure 1 (of which the first 10 are the entire OEIS entry) reproduced exactly

The generating functions of A393920 and A234268 are ALGEBRAIC: summing the source's own a/b recurrences over the second index gives a quadratic equation for B(x,y) with catalytic variable y, and the quadratic method collapses it to 27u³-36u²+8u+1+16xu²+xP = 0 with u = 1-x-x² P, equivalently the rational parametrisation x = (Y-1)/(Y²(3Y-2)), P = Y²(3Y-2)(1+Y-Y²), R = Y²(3Y-2)(5-3Y); the parametrisation satisfies the cubic identically (ring) and the tabulated P satisfies it on x⁰..x¹60 in the kernel

The algebraic certificates that locate the singularity: the numerator of dx/dY is -Y(3Y-4)(2Y-1); its roots 1/2 and 4/3 are the branch points, Y = 0 being a POLE; x(4/3) = 3/32 and x(1/2) = 4 exactly; the RESULTANT res_P(E, dE/dP) of the un-normalised form (leading coefficient -27x⁶) is 27x¹5 (32x-3)³ (x-4) – equivalently the discriminant of the NORMALISED cubic is x⁵ (32x-3)³ (x-4) – a TRIPLE root at 3/32; and 32/3 is a root of the reciprocal quartic 108L⁴-3483L³+37728L²-140288L+32768

The exact growth rate, which the source says it does not know ("the limit probably somewhere around 10.5"): lim Rₙ^(1/n) = lim Pₙ^(1/n) = 32/3 = 10.666..., with Rₙ (768/(25 sqrt(5 pi))) n^(-5/2) (32/3)ⁿ, Pₙ (49152/(2025 sqrt(5 pi))) n^(-5/2) (32/3)ⁿ and Rₙ/Pₙ -> 81/64; this replaces the source's 9ⁿ < Rₙ < 12.75ⁿ by a single constant

A formula for two OEIS entries that carry none: Rₙ = (1/n)tⁿ⁻¹(1+t)²ⁿ⁺¹(1+3t)ⁿ and Pₙ = (1/n)tⁿ⁻¹(1+t)²ⁿ⁺¹(1+3t)ⁿ, and the two-term recurrences 16(n+1)(2n+3)(5n+14)Rₙ + 6(n+4)(2n+7)(5n+9)Rₙ₊₂ = (655n³+4469n²+10090n+7536)Rₙ₊₁ and 16(n+1)(2n+7)Pₙ + 6(n+4)(2n+9)Pₙ₊₂ = (131n²+789n+1162)Pₙ₊₁ — derived from the parametrisation by an exact rational-function ODE certificate, kernel-checked on 519 terms

Controls for the deep package: the 519-term tables extend the landed Rdata/Pdata/Wfull/Vfull exactly, satisfy the source's Lemma P-exact and the lossless exact recurrence on their whole length, satisfy the truncated hypotheses at the levels the new rungs use, are dominated by 2*11ⁿ, and both hypothesis packages have explicit witnesses

Two new rungs of the source's ladder: Rₙ >= (10.3)ⁿ/485 at minimal level L = 325 (lossless) / 331 (the source's own ladder) and Rₙ >= (10.4)ⁿ/1071 at L = 475 / 483, both constants least (484 fails at n = 68, 1070 at n = 96), with the matching bounds Pₙ >= (10.3)ⁿ/623 and Pₙ >= (10.4)ⁿ/1369, also least; and the ceiling, now explained: 10.45, 32/3, 10.7 and 11 are subsolutions at no level L <= 518 (those four theorems are tagged Q21, not this row), and – CONDITIONALLY ON THE GROWTH THEOREM, not by kernel check – no base >= 32/3 ever is